On the cohomology of p-solvable groups.

Suppose V is an irreducible $${\mathbb F}_pG$$ -module where p is a prime and G is a finite p-solvable group. Then by a result of Gaschütz $$\dim _FH^1(G,V)=s_G(V)$$ where $$F=\text {End}_G(V)$$ and $$s_G(V)$$ is the multiplicity of V as a split chief factor in any chief series of G. One also knows that $$\dim _FH^1(G,V)\le \dim _FH^2(G,V)$$ , and the object of the present paper is to explain this inequality in terms of Gaschütz's result. [ABSTRACT FROM AUTHOR]